BT-PEC System under Berry-Taro Private Cross-Key: Topological Protection, Dynamic Equilibrium and Self-Emergence

Within the theoretical framework of BT-PEC (Berry-Taro Private Entanglement Cosmology, Berry-Taro Private Cross-Key Theory), this paper takes the Berry-Taro Private Cross-Key formula as the sole axiomatic starting point. Relying on three inherent native properties of BT-PEC: single anchor, dual vision, and continuous evolution without breakpoint, rigorous algebraic derivation is performed for the tripartite coupled quantum state of anchor-medium information-observable privacy in Hilbert space, yielding the core trace difference identity of the system Ω=tr(BT)−tr(PEC)=0. Under physical premises of finite measurement precision, Bekenstein gravitational entropy bound and mutual independence of three degrees of freedom, all physically distinguishable quantum states of the BT-PEC system can be embedded into the 3-dimensional integer lattice Z^3. Closed topological orbits are generated via orientation-preserving automorphism of the SL(3,Z) unimodular group. Persistent homology analysis gives the stable topological invariants of the system: b0=1, b1=3, b2=1, χ=−1. Based on the above axiomatic system and topological structure, four intrinsic ontological features of the BT-PEC system are strictly derived: non-disclosable, non-replicable, non-tamperable, and indestructible. An N+1-layer nested topological security architecture is constructed on the quotient space Z^3/SL(3,Z). An endogenous gradient backreaction perturbation response mechanism and orbital steady-state dynamic equilibrium mechanism are established. The self-evolution iteration rule and local structure emergence mechanism of the BT-PEC system under global topological conservation are illustrated. This paper strictly distinguishes pure algebraic axiomatic theorems and physically conditional inferences, and establishes a complete system of falsifiability criteria, providing a novel axiom-topology integrated theoretical model for complex quantum information systems, topological protection systems and self-evolving dynamical systems.